NETWORKS AND HETEROGENEOUS MEDIA doi:10.3934/nhm.2019026c©American Institute of Mathematical SciencesVolume 14, Number 4, December 2019 pp. 659–676

WELL-BALANCED SCHEME FOR GAS-FLOW IN PIPELINE

NETWORKS

Yogiraj Mantri∗, Michael Herty and Sebastian Noelle

Institut fur Geometrie und Praktische MathematikRWTH Aachen University

Templergraben 55, 52056 Aachen, Germany

(Communicated by Benedetto Piccoli)

Abstract. Gas flow through pipeline networks can be described using 2 × 2hyperbolic balance laws along with coupling conditions at nodes. The numer-

ical solution at steady state is highly sensitive to these coupling conditions

and also to the balance between flux and source terms within the pipes. Toavoid spurious oscillations for near equilibrium flows, it is essential to design

well-balanced schemes. Recently Chertock, Herty & Ozcan[11] introduced a

well-balanced method for general 2× 2 systems of balance laws. In this paper,we simplify and extend this approach to a network of pipes. We prove well-

balancing for different coupling conditions and for compressors stations, and

demonstrate the advantage of the scheme by numerical experiments.

1. Introduction. The study of mathematical models for gas flow in pipe networkshas recently gained interest in the mathematical community, see e.g. [3, 4, 9, 12, 27].While in the engineering literature [31] the topic has been discussed some decadesago, a complete mathematical theory has only emerged recently, see e.g. [16] for theEuler system on networks, [14] for the p–system on networks and [8] for a recentreview article on general mathematical models on networks. Depending on thescale of phenomena of interest, different mathematical models for gas flow mightbe useful. A complete hierarchy of fluid–dynamic models has been developed anddiscussed in [9]. Therein, typical flow rates and pressure conditions are given andit is shown that a steady state algebraic model can be sufficient to describe averagestates in a gas network. Models based on an asymptotic expansion of the pressuremay lead to further improvements in case of typical, slowly varying, temporal flowpatterns [23, 18]. If a finer resolution of the spatial and temporal dynamics isrequired, the isothermal Euler equations (1) provide a suitable model [3]. Here wefocus on schemes which capture both steady states and small temporal and spatialperturbations.

Schemes which preserve a steady state exactly are called well-balanced, and theirdevelopment is a lively topic in the field of hyperbolic balance laws, see e.g. themonograph [32]. Usually, these schemes use specific knowledge of an equilibriumstate. As a consequence, well-balanced schemes for still water such as [1, 29], orfor moving water, [30], or for wet-dry fronts such as [6, 10], which all approximate

2010 Mathematics Subject Classification. Primary: 35L60, 35L65, 76M12.

Key words and phrases. Well-balanced schemes, hyperbolic balance laws, flows in networks.∗ Corresponding author: Yogiraj Mantri.

659

660 YOGIRAJ MANTRI, MICHAEL HERTY AND SEBASTIAN NOELLE

solutions to the shallow water equations, use different discretization techniques. Aunified approach to well-balancing in one space dimension was recently proposedby Chertock, Herty and Ozcan [11], who integrate the source term in space andsubtract it from the numerical flux. This results in a new set of so-called equilibriumvariables. The equilibrium variables are then reconstructed, and their values at thecell interfaces are used to compute numerical fluxes, which involve a new equilibriumlimiter. Chertock et al. demonstrated the feasibility of this approach using a secondorder scheme for subsonic gas flow in a pipe with wall friction.

To the best of our knowledge, there is currently no well-balanced scheme forhyperbolic flows on networks. Here spurious oscillations may not only be causedby an imbalance of numerical fluxes and source terms, but also by discretizationerrors at junctions and compressors. In the present paper, we extend the equilibriumvariable approach and develop a second order well-balanced scheme on a network.We also simplify the numerical fluxes introduced in [11] by removing the equilibriumlimiter. High-order schemes for gas networks have been introduced only recentlyin [2, 7, 28]. A challenging question would be to extend our new well-balancingmethod to these more accurate schemes.

In the following we introduce the mathematical model for the temporal andspatial dynamics of gas flow in pipe networks. For simplicity, we study a singlenode x = xo where M pipes meet. The flow within each pipe i = 1 . . .M isgoverned by the isothermal Euler equations

(Ui)t + F (Ui)x = S(Ui) (1)

with conservative variables Ui, flux F (Ui) and source S(Ui) given by

Ui =

[ρiqi

], F (Ui) =

[qi

q2iρi

+ p(ρi)

], S(Ui) =

[0

− fg,i2Di

qi|qi|ρi

]. (2)

Here ρi, qi and p(ρi) are the density, momentum, and pressure of the gas, fg,i is thefriction factor and Di the diameter of pipe i. The pressure of the gas is given byGamma-law,

p(ρ) = κργ for γ ≥ 1. (3)

For the numerical tests we focus on the special case of isothermal pressure,

p(ρ) = ρRT = a2ρ, (4)

with speed of sound a > 0. However the well-balanced scheme discussed in thefollowing sections, including the proofs of Theorems 3.1 and 4.1 are valid for anypressure given by Gamma-law (3). We complete (1) with initial conditions withinand boundary conditions at the ends of the pipes. The boundary conditions at anode of multiple pipes are implicitly given by coupling conditions [3, 35], which takethe form of M nonlinear algebraic equations involving traces Ui of the conservedvariables at node xo. We write them in the general form

φ(U1, U2, ..., UM ) = 0, φ : R2M → RM . (5)

In [12, 13, 35] general conditions are identified which guarantee a well–posed problemfor initial data with suitable small total variation. If at a node the conservation ofmass and equality of adjacent pressures is assumed, then existence, uniqueness andcontinuous dependence on the initial data for the p–system was shown in [15].

WELL-BALANCED SCHEME FOR FLOWS IN NETWORKS 661

For pipes with a junction, a flow which is steady within each pipe, and fulfillsthe coupling conditions at the junction is called a steady state [22]. The couplingconditions are further detailed in Sections 2 and 3.

O U3(x)

U1(x)

U2(x)

U∗1U∗3

U∗2

Uo1

Uo2

Uo3

Figure 1. Intersection of three pipes at junction O. Right-Zoomed view of the junction with old traces Uoi and new tracesU∗i given in Section 2

2. Review of coupling conditions for the p–system. Our construction of well-balanced schemes is based upon analytical results for the isothermal Euler equationswhich have already been established (see e.g. [3, 15] and the references therein).For completeness, and to fix the notation, we would like to give a brief summary.

When we study the coupling condition, we set the source terms S(Ui) to zero.This is based on the heuristic assumption that wall friction can be neglected atthe instance of interaction at the node. It can also be justified rigorously for thesemi-discrete scheme (38). The eigenvalues for the homogeneous 2 × 2 system (1)are λ1 = q

ρ − a and λ2 = qρ + a. We assume that all states are subsonic, i.e.,

λ1(Ui) < 0 < λ2(Ui) for i = 1 . . .M. (6)

This assumption is satisfied for typical gas flow conditions in high–pressure gastransmission systems. We denote the set of all incoming (respectively outgoing)pipes by I− (respectively I+). For i ∈ I−, we parametrize the incoming pipes byx ∈ Ωi := (−∞, xo). Similarly, we parametrize outgoing pipes j ∈ I+ by x ∈ Ωj :=(xo,∞). Let us fix a time to ≥ 0. It is important to note that there are 2M differenttraces at the node (xo, to). We denote them by,

Uoi := limx↑xo

limt↓to

Ui(x, t) for i ∈ I−, (7)

U∗i := limt↓to

limx↑xo

Ui(x, t) for i ∈ I−, (8)

Uoj := limx↓xo

limt↓to

Uj(x, t) for j ∈ I+. (9)

U∗j := limt↓to

limx↓xo

Uj(x, t) for j ∈ I+, (10)

662 YOGIRAJ MANTRI, MICHAEL HERTY AND SEBASTIAN NOELLE

Note that the limits are exchanged in (8) and (7) (respectively (10) and (9)), soUoi and Uoj are limits along the x-axis. We call them the old traces. The states U∗iand U∗j are limits along the t axis, and we call them the new traces (see Figure 1).

The construction of the coupling conditions starts by connecting, within eachpipe, the old with the new trace by a Lax curve entering the pipe. For an incom-ing pipe, this will be a curve U i(σi) of the first family with left state Ul := Uoi .According to [17], this curve for γ = 1 is given by

1−R : U i(σi) := ρl eσi

[1

ul − aσi

]for σi ≤ 0,

1− S : U i(σi) := ρl(1 + σi)

[1

ul − aσi√1+σi

]for σi > 0.

(11)

Analogously, for an outgoing pipe, we use curves of the second family with rightstate Ur := Uoj , which are given by

2−R : U j(σj) := ρr eσj

[1

ur + aσj

]for σj ≤ 0,

2− S : U j(σj) := ρr(1 + σj)

[1

ur +aσj√1+σj

]for σj > 0.

(12)

We refer to [17] for case γ > 1. The Lax curves 1-R and 1-S(respectively 2-R and2-S) have C2 continuity at the point Ul(respectively Ur).

The parameters σi, i = 1 . . .M will be determined from the M coupling condi-tions

φ(σ1, . . . , σM ) := φ(U1(σ1), . . . , UM (σM )) = 0. (13)

Now we set

U∗i := U i(σi). (14)

By construction, the new traces satisfy the coupling conditions (5). Note thatthe number of unknowns in (13) are halved compared to (5) as each conservativevariable, Ui ∈ R2 can be written in terms of single parameter σi ∈ R.

So far, we have reviewed the general framework which was established and appliedin [3, 15, 33]. In the following we focus on a particular coupling condition for whichwe design a well-balanced scheme. Let Ai = π

4D2i be the area of the cross section

of pipe i. The default coupling condition requires that the total incoming mass fluxat each node xo equals the total outgoing mass flux,∑

i∈I−Aiq∗i =

∑j∈I+

Ajq∗j , (15)

since mass should not be accumulated or lost at the junction. Various approacheshave been studied in order to model the other (M − 1) coupling conditions. Aseemingly obvious choice would be conservation of momentum. However as mo-mentum is a vector quantity it is difficult to describe the conservation of momen-tum in a one-dimensional model as the junctions of the pipe are three-dimensional.Multi-dimensional approaches considering a 2D node for a 1D flow have been dis-cussed in [24, 5]. Coupling conditions based on enthalpy have also been studied in[3, 27, 35, 34]. In general these coupling conditions can be represented in the form,

h(ρ∗i , q∗i ) = h∗(to) for all i ∈ I− ∪ I+. (16)

WELL-BALANCED SCHEME FOR FLOWS IN NETWORKS 663

In [35], Reigstad et al. showed that the coupling conditions are well-posed if thefunction h is monotonic with respect to the parameter σ of the Lax curves,

d

dσh(ρi(σ), qi(σ)) > 0 for all i ∈ I− ∪ I+. (17)

Our analytical results are obtained under the general coupling condition (16) (re-spectively (31)), together with the monotonicity condition (17). In the numericalexperiments, we use the following pressure-coupling condition: we require that thetraces of the pressures should take one and the same value p∗ = p∗(to) at the t-axis,

p(ρ∗i ) = p∗(to) for all i ∈ I− ∪ I+. (18)

This condition is common in the engineering literature, and for a certain regime itis indeed a suitable approximation of the two-dimensional situation [24].

Thus the coupling function (5) at a junction reads

φ(U1, . . . , UM ) =

∑i∈I− Aiqi −

∑j∈I+ Ajqj

p(ρ2)− p(ρ1)...

p(ρM )− p(ρM−1)

. (19)

We now turn to a junction which models a compressor between the incoming pipei = 1 and the outgoing pipe i = 2, both of the same diameter. The couplingconditions for the new traces are

q∗1 = q∗2 , p(ρ∗2) = CRp(ρ∗1). (20)

Here CR ≥ 1 is the compression ratio. It is usually time–dependent, and we considerit to be a given, external quantity. Thus the coupling function for the compressorbecomes

φ(U1, . . . , UM ) =

[q2 − q1

p(ρ2)− CRp(ρ1)

]. (21)

Summarizing, the analytical problem at the nodes is to connect the old tracesUoi within each pipe to a new trace U∗i along the incoming Lax curve in such a waythat the new traces satisfy the coupling conditions across the node. It was provenin [20, 12, 21] that this problem has a unique solution.

If the old traces are subsonic, and their variation is small enough, then the newtraces will be subsonic as well. The new traces serve as initial data in the Riemannsolver which determines the numerical flux.

In this paper, we show the analytical results of well-balancing for the case ofmultiple pipes meeting at the junction. However the scheme is valid for the case ofcompressor as well, as can be seen from the numerical results in Section 5.2.

3. Coupling conditions in terms of equilibrium variables. The difficulty inpreserving steady states is that the divergence of the conservative fluxes is approx-imated by a flux-difference, while the source is usually integrated by a quadratureover the cell. If this is not tuned carefully, the equilibrium state is not maintained,and spurious oscillations may be created. Chertock, Herty and Ozcan [11] resolvedthis difficulty for one-dimensional balance laws by integrating the source term andhence writing it in conservative form. They applied this approach to the Cauchyproblem for 2×2 balance laws. Here we extend their method to a node in a network.Equation (1) can be stated as

664 YOGIRAJ MANTRI, MICHAEL HERTY AND SEBASTIAN NOELLE

(ρi)t + (Ki)x = 0, (qi)t + (Li)x = 0 (22)

where the flux variable,

Vi(Ui, Ri) =

[Ki

Li

]= F (Ui) +

[0Ri

](23)

and fluxes Ki, Li and an integrated source term Ri is given by

Ki := qi, Li :=q2i

ρi+ p(ρi) +Ri(x), Ri(x) :=

∫ x

xi

fg,i2Di

qi|qi|ρi

dx. (24)

The point xi belongs to Ωi and is arbitrary but fixed. Later on, we choose xi = xofor all i. We call (K,L) the equilibrium variables, since they are constant for steadystates. Let us consider the general pressure law (3). Given the integrated sourceterm Ri and the equilibrium variables Vi = (Ki, Li) we now solve equation (24)for the conservative variables (ρi, qi). Omitting the subscripts of the pipes in thefollowing calculation, we rewrite (24) as

ρp− ρ(L−R) = −K2.

In other words, our task is to find the non-negative real roots of

ϕ(ρ) = c (25)

with

ϕ(ρ) := ργ+1 − bρ, b :=L−Rκ

≥ 0, c := −K2

κ≤ 0. (26)

For ρ ≥ 0 the function ϕ is convex, has roots ρ0 = 0 and ρ1 = b1/γ and a criticalpoint in

ρ∗ =( b

γ + 1

)1/γ

with corresponding minimum value

ϕ∗ = −γ( b

γ + 1

)1+1/γ

.

An elementary calculation shows that if K,L and R belong to a sonic state (ρs, us),then ρ∗ = ρs. Thus, if c = ϕ∗, then ρ∗ is a double root of (25). Else, if 0 ≥ c > ϕ∗,we have two real roots ρ±, with

0 ≤ ρ− < ρ∗ < ρ+ < ρ1.

We find the supersonic root ρ− by a Newton iteration with initial value ρ0, and thesubsonic root ρ+ using the initial value ρ1. By convexity of ϕ, the convergence isquadratic. In particular, if γ = 1, then the subsonic root in the ith pipe is given by

ρi(Vi, Ri) =Li −Ri +

√(Li −Ri)2 − 4K2

i a2

2a2, (27)

and hence

Pi(Vi, Ri) = a2ρi(Vi, Ri). (28)

Note that Ri appears as a parameter in (27). Similar to the discussion in theprevious section the conditions are stated for the traces of the equilibrium variables

WELL-BALANCED SCHEME FOR FLOWS IN NETWORKS 665

at xo. The dependence on xo is omitted for readability.∑i∈I−

AiK∗i =

∑j∈I+

AjK∗j , (29)

P (K∗i , L∗i ) = p∗ for all i ∈ I±, (30)

or more generally

H(K∗i , L∗i ) = h∗ for all i ∈ I± (31)

where H could be any any quantity such as momentum flux or Bernoulli invariantas discussed in Section 2. Similarly, the coupling condition for a compressor interms of K and L reads

P (K∗2 , L∗2) = CR P (K∗1 , L

∗1). (32)

For subsonic states, the coupling conditions (29),(30), (32) are equivalent to thecoupling conditions (15),(18), (20).

For steady states, Ki and Li are constant within each pipe, and the couplingconditions are fulfilled at the junction. Evaluating the coupling conditions in termsof the equilibrium variables V = (K,L) and the parameter R is an essential ingre-dient of the well-balancing. Therefore, we rewrite the Lax-curves in terms of theequilibrium variables:

1−R : V i(σi) := ρl eσi

[ul − aσi

(ul − aσi)2 + a2

]+

[0Rl

]for σi ≤ 0,

1− S : V i(σi) := ρl(1 + σi)

ul − aσi√1+σi(

ul − aσi√1+σi

)2

+ a2

+

[0Rl

]for σi > 0.

(33)

Similarly for the admissible boundary states on the pipes i ∈ I+ with given valueRr and ur = qr/ρr, we obtain

2−R : V j(σj) := ρr eσj

[ur + aσj

(ur + aσj)2 + a2

]+

[0Rr

]for σj ≤ 0,

2− S : V j(σj) := ρr(1 + σj)

ur +aσj√1+σj(

ur +aσj√1+σj

)2

+ a2

+

[0Rr

]for σj > 0.

(34)

The equilibrium variables satisfying the coupling condition (29) and (30) aregiven by

K∗i := Ki(σi) and L∗i := Li(σi). (35)

Note that all variables defined along the Lax-curves also depend on the old tracesand the integrated source terms as parameters, e.g. V i(σi) = V i(σi;V

oi , R

oi ). For

given datum Ul, Ur we depict the parameterized wave curves for incoming andoutgoing pipes in the phase space of K and L, respectively, in Figure 2. Fromthe figure we observe that in the subsonic region, the 1–Lax curve is monotonicallydecreasing and 2–Lax curves is monotonically increasing. The following theoremproves that the coupling conditions stated in the variables K and L locally have aunique solution.

Theorem 3.1. Consider a nodal point with |I−| ≥ 1 incoming and |I+| ≥ 1 outgo-

ing adjacent pipes. Suppose that the initial data Ui = (ρi, qi), i ∈ I± are subsonic on

each pipe and fulfill the coupling conditions (15) and (16). Let Vi = (Ki, Li), i ∈ I±

be the corresponding equilibrium variables, with integrated source terms Ri.

666 YOGIRAJ MANTRI, MICHAEL HERTY AND SEBASTIAN NOELLE

(a) Incoming pipes, i ∈ I− (b) Outgoing pipes, i ∈ I+

Figure 2. Phase plot in terms of equilibrium variables with initialstate V oi = (0.1, 0.4)T

Then there exists an open neighborhood V ⊂ R2M×M of (V , R) := (Vi, Ri)i∈I±such that for any old trace (V o, Ro) ∈ V there exists a unique new trace V ∗ such that(V ∗, Ro) ∈ V fulfill the coupling conditions (29) and (30). Moreover, V ∗i is con-nected to V oi by an incoming Lax curve along the respective pipe. The neighborhoodcan be chosen sufficiently small, such that the corresponding states are subsonic.

Proof. Denote by M = |I−|+ |I+| the total number of connected pipes. For V :=(Vi)i∈I± := (Ki, Li)i∈I± the coupling conditions (29) and (30) are given by thefunction Ψ : R2M × R2M × RM → RM .

Ψ(V, (V o, Ro)) :=

∑i∈I− AiKi −

∑j∈I+ AjKj

H(V1)−H(V2)...

H(VM−1)−H(VM )

(36)

where H is a given coupling condition of the form (31). By assumption we have

Ψ(V , (V , R)) = 0. Now, we define

Ψ(σ, (V o, Ro)) := Ψ(V (σ), (V o, Ro)) : RM × R2M × RM → RM

whereV (σ) := (V i(σi))i∈I±

and the components of V i(σi) are given by equation (33) and equation (34), respec-tively. Further, σ = (σi)i∈I± . Next, we compute the determinant of DσΨ(σ, (V o,Ro)) at σ = 0. We have for i ∈ I− and j ∈ I+

DσΨ =

A1dK1

dσ1. . . A|I−|

dK|I−|dσ|I−|

−A|I−|+jdK|I−|+jdσ|I−|+j

. . . j = 1, ...|I+|dH1

dσ1−dH2

dσ20 . . . . . . 0

0 dH2

dσ2−dH3

dσ3. . . . . . 0

. . .. . .

. . .. . .

0 . . . . . . 0 dHM−1

dσM−1−dHMdσM

and therefore

det(DσΨ) = (−1)M−1∑i∈I−

(Ai

dKidσi

∏k∈I−,k 6=i

dHkdσk

)+(−1)M

∑j∈I+

(Aj

dKjdσj

∏k∈I−,k 6=j

dHkdσk

),

(37)

WELL-BALANCED SCHEME FOR FLOWS IN NETWORKS 667

From equations (33) and (34), we obtain at σi = 0

dKi

dσi(0) = qoi − aρoi < 0,

dLidσi

(0) =(qoi − aρoi )2

ρoi> 0,∀i ∈ I−,

dKi

dσi(0) = qoi + aρoi > 0,

dLidσi

(0) =(qoi + aρoi )

2

ρoi> 0,∀i ∈ I+.

Hence,

sign(det(DσΨ)) = (−1)M∑i

sign(dHi

dσi

).

Thus for any monotonic coupling condition, we see that det(DσΨ) 6= 0. The cou-pling conditions such as pressure, momentum flux , Bernoulli invariant mentionedin Section 2 are all monotonic at the point σi = 0.

For example, for the pressure coupling condition (using H(Ki, Li, Ri) = Pi in(31)), we get

dHi

dσi=

1

2

(dLidσi

+(Li −Ri)dLidσi

− 4a2KidKidσi√

(Li −Ri)2 − 4a2K2i

).

Hence, dHidσi

(σi = 0) = a2ρoi > 0 and therefore detDσΨ(0, (V o, Ro)) 6= 0.Similarly for coupling condition given by continuity of momentum flux,

H(Ki, Li, Ri) = Li −Ri and hence

dHi

dσi(0) =

dLidσi

(0) > 0.

Thus, by the implicit function theorem there exists an open neighborhood V of

V such that for all initial data V o ∈ V there exists σ∗ such that V ∗ := V (σ∗) fulfillsthe coupling conditions, i.e.

Ψ(V ∗, (V o, Ro)) = Ψ(σ∗, (V o, Ro)) = 0.

Since the corresponding state of V in conservative variables is strictly subsonicwe may assume, by possibility decreasing the size of V, that also the conservativevariables corresponding to (V ∗, Ro) are subsonic.

Using the same technique, we can also treat the compressor, because the pressurechanges monotonically along the Lax curves.

Corollary 1. Consider a nodal point with |I−| ≥ 1 incoming and |I+| ≥ 1 outgoingpipes or a compressor connected to 1 incoming and 1 outgoing pipe. Suppose we

have an equilibrium state with subsonic initial data Ui = (ρi, qi), i ∈ I± in eachpipe and fulfilling the pressure coupling conditions (15) and (18) for a junction or

(20) for compressor. Let Vi = (Ki, Li), i ∈ I± be the corresponding equilibrium

variables, with integrated source terms Ri. Then there exists an open neighborhood

V ⊂ R2M×M of (V , R) := (Vi, Ri)i∈I± in the subsonic regime, such that for any oldtrace (V o, Ro) ∈ V there exists a unique new trace V ∗ such that (V ∗, Ro) ∈ V fulfillthe coupling conditions (29) and (30) for a junction or (32) for a compressor.

Proof. The corollary follows from the proof of Theorem 3.1. One can also checkthat the pressure for a general Gamma law is also monotonic, i.e.,

dPidσi

(σi = 0) =γPi

dLidσi− 2γKi

dKidσi

(Piκ

)−1γ

Pi

(γ + 1)Pi − (Li −Ri)> 0.

668 YOGIRAJ MANTRI, MICHAEL HERTY AND SEBASTIAN NOELLE

In case of a compressor the pressure of the gas is multiplied by the compression ratioCR > 0 and hence it does not affect the monotonicity of the coupling condition.

Remark 1. Note that we have omitted the source term when computing the solu-tion along the Lax curves. This is justified by the semi-discrete formulation of thefinite volume scheme in the next section, which implies that we are evaluating thecoupling condition at a set of measure zero in space-time. Since the source term isbounded, it does not contribute to the integral over the cells.

Remark 2. Another possibility is to reformulate system (22) as a system of threeequations in (Ki, Li, Ri) with the equation for Ri given by

∂tRi = 0.

Therefore, the corresponding hyperbolic field has a zero eigenvalue in an indepen-dent subspace. This yields a characteristic boundary at each adjacent pipe. Hencein phase space, at each pipe i any value Ri can be connected along a wave curve toRi. The central wave leads to a contact discontinuity of zero velocity at the nodalpoint. Hence, the trace of Ri at x = xo is independent of Ri.

4. A well-balanced central-upwind scheme for nodal dynamics. We com-pute the evolution of the conservative variables using the second–order centralupwind scheme [26, 25]. The computational domain Ωi is discretized in cells[xj− 1

2, xj+ 1

2] of size ∆x and centered at xj = x + (j − 1

2 )∆x for j = 1, . . . , N .

We choose x such that xN = xo for i ∈ I− and x0 = xo for i ∈ I+. For simplicitythe same number of cells N for all adjacent pipes will be used. The approximatedcell averages at fixed time t are computed as

U ji (t) =1

∆x

∫ xj+1

2

xj− 1

2

Ui(x, t)dx, , i ∈ I±, j = 1, . . . , N.

The evolution of conservative variables, density and momentum using central up-wind scheme [25, 26] reads

dU jidt

= −Vj+1/2i − Vj−1/2

i

∆x(38)

where Vj−1/2i ,Vj+1/2

i are the fluxes across the left and right interface of cell j,respectively. At the junction, the flux is the new trace of the equilibrium variable,

VN+1/2i = V ∗i , i ∈ I−, (39)

V1/2i = V ∗i , i ∈ I+. (40)

The new traces V ∗i are constructed with the help of Theorem 3.1 based on the old

traces V N,Ei , i ∈ I− and V 1,Wi , i ∈ I+. The point values of K and L at the cell

interfaces, i.e., Kj,Ei ,Kj,W

i , Lj,Ei , Lj,Wi , are computed using piecewise linear recon-

struction of Kji and Lji calculated using the cell averages (ρji , q

ji ) using equation

(24). The values Ri are computed using a second–order quadrature rule applied tothe integral starting for example at x = xo with Ri = 0 at each pipe according tothe following equations

R1/2i = R

N+1/2k = 0 ∀i ∈ I+, k ∈ I−,

Rj+1/2i = R

j−1/2i +∆x

fg,i2Di

qji |qji |

ρji, R

j−1/2k = R

j+1/2k + ∆x

fg,k2Dk

qjk|qjk|

ρjk.

WELL-BALANCED SCHEME FOR FLOWS IN NETWORKS 669

The equilibrium variables W ∈ K,L at the right and left boundaries of cell j canbe calculated as,

W j,Ei = W j

i +∆x

2(Wx)ji , W

j,Wi = W j

i −∆x

2(Wx)ji (41)

with numerical derivatives

(Wx)ji =

W j+1i −W j

i

∆x , j = 1

W ji −W

j−1i

∆x , j = N

minmod(θW j+1i −W j

i

∆x ,W j+1i −W j−1

i

2∆x , θW ji −W

j−1i

∆x

), otherwise,

(42)

θ ∈ [1, 2] and minmod limiter

minmod(w1, w2, . . . , wn) =

min(w1, w2, . . . , wn) if wi > 0, ∀imax(w1, w2, . . . , wn) if wi < 0, ∀i0 otherwise

(43)

For interior interfaces, we may use any conservative numerical flux functions whosenumerical diffusion vanishes at equilibrium states. Here we choose the central up-wind flux,

Vj+1/2i =

aj+1/2i,+ V j,Ei − aj+1/2

i,− V j+1,Wi

aj+1/2i,+ − aj+1/2

i,−

+ αj+1/2i (U j+1,W

i − U j,Ei ), (44)

where aj+1/2i,± are the maximum and minimum eigenvalues of the Jacobian, i.e.,

aj+1/2i,+ = max

(λ(U j,Ei ), λ(U j+1,W

i ), 0), a

j+1/2i,− = min

(λ(U j,Ei ), λ(U j+1,W

i ), 0)

(45)

and αj+1/2i is the local diffusion computed as α

j+1/2i =

aj+1/2i,+ a

j+1/2i,−

aj+1/2i,+ −aj+1/2

i,−. The conser-

vative variables U j+1,Wi , U j,Ei can be computed from the corresponding equilibrium

variables V j+1,Wi , V j,Ei and the integral term R

j+1/2i using equation (27).

Remark 3. Note that in [11] an additional limiter was introduced to suppress thenumerical viscosity in (44) at equilibrium and assure well-balancing:in particular,

the numerical viscosity αj+1/2i was multiplied with a factor

H(φ) =(Cφ)m

1 + (Cφ)m(46)

where φ was given by|Kj+1−Kj |

∆x|Ω|

maxKj ,Kj+1 for the mass equation and analogously

with K replaced by L for the momentum equation.

The following theorem shows that well-balancing is already assured by the con-

tinuity of the integrated source terms Rj+1/2i at equilibrium.

Theorem 4.1. The numerical scheme given by (38) and flux defined by (44) pre-serves the steady state across a node of M adjacent pipes and coupling conditionsgiven by (29) and (31).

670 YOGIRAJ MANTRI, MICHAEL HERTY AND SEBASTIAN NOELLE

Proof. Consider steady state (V , R) := (Vi, Ri)i∈I± . Then all numerical derivativesdefined in (42) vanish at equilibrium.

Since the equilibrium variables V j+1,Wi = V j,Ei = Vi and the integral term R

j+1/2i

are constant across the cell interface, we see that the conservative variables at the

cell interfaces are also continuous at equilibrium i.e. U j+1,Wi = U j,Ei . Thus the

numerical fluxes are given by Vj+1/2i = Vi for all j = 2, . . . , N − 1.

At the nodal point the flux variables satisfy the coupling conditions (29) and(30). Then, the boundary data K∗i and L∗i are obtained according to Theorem 3.1.

Since the states are unique we obtain Kji = K∗i = Ki and Lji = L∗i = Li and hence

the boundary fluxes for each pipe at the junction are VN+1/2i = VN−1/2

i = (Ki, Li)T

for incoming pipes i ∈ I− and V1/2j = V3/2

j = (Kj , Lj)T for outgoing pipes j ∈ I+.

Hence, the scheme is well-balanced across the node.

Data: Given discretized initial conditions U ji (0) = Ui(x, 0)while terminal time not reached do

Compute equilibrium variables (Kji , L

ji , R

ji ) by (24) ;

Reconstruct the values of K and L at the cell interface by (41) ;Solve the coupling conditions,(29),(30) to find K∗i , L

∗i Calculate

conservative variables (ρ, q) at the cell interface by equations (27) ;Calculate fluxes (44) for interior cell boundaries and use K∗i , L

∗i at

junction ;

Compute the time step ∆t = CFL∆x

maxi,j |λji |where λji is the maximal

eigenvalue of the Jacobian in cell j and pipe i;Evolve the conservative cell averages (38).

end

Some remarks are in order. The algorithm uses the same time step for all ad-jacent pipes. This is not necessary but simplifies the computation of the couplingcondition. Also, the algorithm is second–order in the pipe but it may reduce to firstorder at the coupling condition. The algorithm can be extended to second–orderacross the nodal point using techniques presented in [2]. However, note that thesteady state is constant and therefore the scheme preserves the steady state to anyorder across the nodal point.

5. Numerical tests. In this section, we test the well-balanced(WB) scheme withnumerical examples for steady state and near steady state flows. The resultsof this WB method have been compared with a second order non well-balancedmethod(NWB). The NWB scheme is given by,

dU jidt

= −Fj+1/2i −F j−1/2

i

∆x+ Sji (47)

where F is the HLL flux given by,

F j+1/2i =

aj+1/2i,+ F (U j,Ei )− aj+1/2

i,− F (U j+1,Wi )

aj+1/2i,+ − aj+1/2

i,−

+ αj+1/2i (U j+1,W

i − U j,Ei )

where the flux terms are as defined in (2) and Sji is the source term given in (2) at

the point U ji . The coupling conditions (15), (18) are used to calculate the densityand momentum at a node. The coupling conditions at the node are solved withNewton’s method for both WB and NWB schemes with initial guess for Newton’s

WELL-BALANCED SCHEME FOR FLOWS IN NETWORKS 671

iteration given by the solution in the pipe at the node, V N,Ei /UN,Ei ∀i ∈ I− and

V 1,Wi /U1,W

i ∀i ∈ I+ for WB/ NWB schemes respectively.We run the tests at CFL number=0.4 and minmod parameter, θ = 1. All the

pipes in the examples have been considered to be of same diameter and friction

factor,fg2D = 1 and the speed of sound for the gas, a = 1. We test the scheme for

several well-balanced flows at junctions and compressors, as well as perturbationsof such steady states.

5.1. Steady state at a node. In the first example, we study the WB scheme forsteady state at a node with three types of pipe combinations–1 incoming and 1outgoing pipes; 1 incoming and 2 outgoing pipe; and 2 incoming and 1 outgoingpipe. The initial conditions are selected in such a way that the node is at steadystate with the equilibrium variables constant in each pipe and satisfying the couplingconditions at the node.

The initial condition for first case with 1 incoming and 1 outgoing pipe areK1 = K2 = 0.15 and p∗ = 0.332 corresponding to L1 = L2 = 0.4. Similarly forthe second case, of 1 incoming and 2 outgoing pipe, K1 = 0.15,K2 = K3 = 0.075and p∗ = 0.332 or L1 = 0.4, L2 = L3 = 0.3492; and for 2 incoming and 1 outgoingpipes, K3 = 0.15,K1 = K2 = 0.075 and p∗ = 0.332 or L3 = 0.4, L1 = L2 = 0.3492.

The L-1 error for the three cases are given in Table 1,

1 Incoming, 1 Outgoing 1 Incoming, 2 Outgoing 2 Incoming, 1 Outgoing

No. of cellsin each pipe

L1-errorfor variable WB NWB WB NWB WB NWB

50 K 2.83x10−17 6.19x10−7 6.91x10−17 3.78x10−7 9.02x10−17 3.45x10−7

L 3.44x10−17 9.48x10−7 5.16x10−17 3.57x10−7 9.21x10−17 7.38x10−7

100 K 3.95x10−17 1.56x10−7 8.12x10−17 9.63x10−8 8.60x10−17 8.67x10−8

L 4.86x10−17 2.43x10−7 7.38x10−17 8.94x10−8 8.24x10−17 1.87x10−7

200 K 5.11x10−17 3.88x10−8 8.69x10−17 2.62x10−8 1.04x10−16 2.69x10−8

L 5.85x10−17 6.13x10−8 7.06x10−17 2.32x10−8 9.49x10−17 5.03x10−8

Table 1. Comparison of L-1 errors between well-balanced(WB)and non well-balanced(NWB) scheme at steady state for a junctionat time T=1

As can be seen from the results in Table 1, the L-1 error ||K−K|| and ||L−L|| isaccurate up to machine precision using the WB scheme,whereas it is of the order of10−7 to 10−8 with the NWB scheme. We can also note that the coupling conditionsin terms of (K,L) converge quickly using Newton’s method and do not affect thewell-balancing property of the scheme at the node.

5.2. Steady state with a compressor. In the second example, we study thewell-balancing at steady state for compressor connecting two pipes with compres-sion ratios CR = 1.5, 2, 2.5. The initial conditions are selected in a way that thecompressor is at steady state for time,T=0. The momentum in the two pipes isgiven by K1 = K2 = 0.15 and pressure is given by p∗1 = 0.332 and p∗2 = CRp∗1. TheL1 errors using the WB and NWB scheme are given in Table 2.

Similar to the first example, we see that the L1 errors using the WB scheme areaccurate up to machine precision. Also the coupling conditions for the compressordo not affect the well-balancing of the scheme.

672 YOGIRAJ MANTRI, MICHAEL HERTY AND SEBASTIAN NOELLE

CR=1.5 CR=2.0 CR=2.5

No. of cellsin each pipe

L1-errorfor variable WB NWB WB NWB WB NWB

50 K 1.11x10−17 4.16x10−7 5.30x10−17 3.78x10−7 1.97x10−17 3.77x10−7

L 2.66x10−17 4.00x10−7 5.38x10−17 3.57x10−7 1.39x10−17 3.54x10−7

100 K 2.90x10−17 1.05x10−7 7.28x10−17 9.63x10−8 4.22x10−17 9.68x10−8

L 4.08x10−17 1.01x10−7 7.24x10−17 8.94x10−8 4.66x10−17 8.89x10−7

200 K 4.26x10−17 2.64x10−8 8.15x10−17 2.62x10−8 5.02x10−17 2.84x10−8

L 4.69x10−17 2.53x10−8 7.45x10−17 2.32x10−8 5.76x10−17 2.59x10−8

Table 2. Comparison of L-1 errors between well-balanced(WB)and non well-balanced(NWB) scheme at steady state with a com-pressor at different compression ratios at time T=1

5.3. Perturbations to steady state for a node. From the first two examples,we can see that the WB scheme preserves steady state. We now compare the WBand NWB scheme for initial conditions given by perturbations to momentum atsteady state. The initial conditions for the perturbed state are given by,

Ki(x) = Ki + ηie−100(x−x0)2 , Li = Li ∀i = 1, 2 . . .M (48)

where Ki and Li are constant steady state equilibrium variables in the two pipesand ηi is the magnitude of perturbation at the node.

At first, we consider a node connecting two pipes. The equilibrium variables for

this case are given by, Ki = 0.15 and Li = 0.4. At first we consider perturbation ofηi = 10−3 at the junction. The momentum at time T=0.2 are as shown in Figure3,

(a) First Pipe (b) Second Pipe

Figure 3. Momentum for perturbation of order 10−3 for a nodeconnected to two pipes

We can see from the results that both the WB and NWB schemes provide similarsolutions for the perturbation of order 10−3 at the node. We now reduce thisperturbation to ηi = 10−6.

From Figure 4 we can see that the NWB scheme develops oscillations for N=100when the perturbation is of order 10−6. The perturbation is resolved better for afiner grid with N=500 per pipe. However in the case of WB method, the scheme isable to capture the perturbations well even for a coarser grid of N=100 per pipe.

We now do a similar test for a node connected to 1 incoming and 2 outgoing

pipes. The equilibrium states are given by, K1 = 0.15, K2 = K3 = 0.075 and L1 =

0.4, L2 = L3 = 0.3492. We also run this simulation for two perturbations of order10−3 and 10−6 up to a time T=0.2. Figure 5 shows the result for momentum with

WELL-BALANCED SCHEME FOR FLOWS IN NETWORKS 673

(a) First Pipe (b) Second pipe

Figure 4. Momentum for perturbation of order 10−6 for a nodeconnected to two pipes

η∗1 = 10−3, η∗2 = η∗3 = 0.5x10−3 and Figure 6 for η∗1 = 10−6, η∗2 = η∗3 = 0.5x10−6

respectively.

(a) First pipe (b) Second pipe

(c) Third pipe

Figure 5. Momentum for perturbation of order 10−3 for a nodeconnected to one incoming and two outgoing pipes

We see from the results that even for the perturbations of order 10−3, the resultsfrom NWB scheme are unstable when there is a sharp increase in momentum. Theresults of NWB scheme are even more oscillatory when the perturbations are oforder 10−6. Further even with a finer resolution, we can see a spike in the regionwhere there is a jump in momentum. However, these issues are resolved with theWB scheme. The results of WB scheme with coarser grid are a bit more diffusivethan the finer grid, but there are no instabilities arising in the solution.

5.4. Shock at the node. So far we have tested the scheme for smooth near-equilibrium solutions. Now we consider a discontinuity at a junction of 1 incomingand 2 outgoing pipes, i.e. ρ1 = 5, ρ2 = 4, ρ3 = 3 and q1 = q2 = q3 = 1. This resultsin a rarefaction and two shocks moving into the pipes, so the solution is far fromequilibrium. The boundary condition is given by momentum at the inlet, qin = 1and density at outlet, ρ2,out = 4, ρ3,out = 3. The solution at time T = 0.1, 0.25 withWB and NWB scheme are shown in Figure 7 and Figure 8 respectively.

674 YOGIRAJ MANTRI, MICHAEL HERTY AND SEBASTIAN NOELLE

(a) First pipe (b) Second pipe

(c) Third pipe

Figure 6. Momentum for perturbation of order 10−6 for a nodeconnected to one incoming and two outgoing pipes

(a) Density (b) Momentum

Figure 7. Conservative variables, ρ, q at T=0.1 in pipes 1, 2, 3with WB and NWB scheme

(a) Density (b) Momentum

Figure 8. Conservative variables, ρ, q at T=0.25 in pipes 1, 2, 3with WB and NWB scheme

From the above test, we can see that the scheme is able to capture shocks. Thesolution shows a 1-wave moving towards left into the first pipe from the junctionand a 2-wave towards right into the second and third pipes. Similar solutions wereobtained by Egger[19] using a conservative mixed finite element method with a

WELL-BALANCED SCHEME FOR FLOWS IN NETWORKS 675

stagnation enthalpy coupling condition. In Figures 7 and 8, we use the classical,non-well -balanced scheme (using U -variables) as reference and observe that thesolutions are nearly identical. We can also note that the solution satisfies thecoupling conditions at the node i.e. the pressure is constant at the node and sum ofmomentum of second and third pipe is equal to the momentum of first pipe. Notethat the Newton iteration failed for this example when using the H-limiter (46).

6. Conclusion. In this paper we have extended the recent, equilibrium variablebased approach to well-balancing, developed by Chertock, Herty and Ozcan[11] forone-dimensional systems, to a network of gas pipelines with friction. In particular westudied intersections of pipes at a node and compressors within a pipeline network.We prove well-posedness and well-balancing of the new scheme. For compressorsand for junctions of three pipes, numerical experiments demonstrate that equilibriaare resolved up to machine accuracy. Most interestingly, near equilibrium flowsare resolved robustly and accurately, even in cases where a standard non-balancedscheme fails. For flows away from equilibrium, including shocks emanating from ajunction, the scheme is as good as a standard scheme using conservative instead ofequilibrium variables.

Acknowledgments. This work has been supported by HE5386/13–15, BMBFENets Project, and DFG Research Training Group 2326: Energy, Entropy, andDissipative Dynamics, RWTH Aachen University.

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Received June 2018; 1st revision February 2019; 2nd revision May 2019.

E-mail address: mantri@eddy.rwth-aachen.de

E-mail address: herty@igpm.rwth-aachen.de

E-mail address: noelle@igpm.rwth-aachen.de